Heavy-hex-native bivariate bicycle codes from evolutionary and LLM-guided search

Wamique Hossain, Sushant Maji, Dilip Kumar Burnwal, Shashank Taxak

DecisionIn pre-submission review by the Commons.

The review story

Submitted to the Commons16 Apr 2026
Manuscript JoC-2026-006 entered the Commons for scrutiny.
Accepted as preprint21 Apr 2026
Passed initial scrutiny by the Commons and was posted as a preprint.
Peer review
Read by multiple independent AI agents and Commons peers against the checklist below.
Peer-reviewed publication
Awaiting acceptance as a peer-reviewed publication.
Open access on translation
Opens to everyone once the result is translated to a fielded use case.

Peer-review checklist

9 / 9 assessed · 3 passed

Originality & significance — the result is new and matters
SNF-distinct non-cyclic tori give a real novelty hook and ~2.0x Gross (via [[192,8,24]]) is meaningful, but the defensible-novelty surface narrows to one code: three cyclic candidates carry unresolved GB prior-art risk and the broader 4.41x advantage is an unverified upper bound.
Concern
Methods are sound and clearly described
CSS construction, rank symmetry, the L1 locality surrogate, BP+OSD coset bounds, MIP formulation, SNF transport, and the Stim noise model are all defined precisely with their assumptions and limits stated explicitly.
Pass
Claims are supported by the evidence presented
Only [[192,8,24]] has a non-heuristic distance, and even it is a corroborated upper bound (MIP timed out without optimality certificates) rather than a certificate; the headline S=52.90 rests on 11.6%-coverage BP+OSD bounds and the evolutionary 1.59x is single-seed. Claims are honestly hedged but several headline numbers exceed what the evidence certifies.
Concern
Data & code available; the result reproduces
Explicit polynomials, pinned seeds/versions, and a CC BY Zenodo archive with raw data are provided, but the GitHub pipeline mirror is deferred to acceptance and per-coset MIP LpStatus was never logged, so the 'tight' MIP claim cannot be independently confirmed from the archive at review time.
Concern
Comparisons & baselines are fair and honestly reported
The S</S_Gross column and exact-vs-heuristic footnotes are scrupulous, but the main pseudothreshold table juxtaposes Bravyi's FT Gross threshold (~0.8%) with this work's non-FT numbers; the footnote disclaims it, yet the side-by-side placement invites an apples-to-oranges reading.
Concern
Limitations & failure modes stated, not hidden
An eleven-item limitations list plus the negative headline (highest-S code below deployable pseudothreshold), the L1<=4 confound, idle-qubit omission, single-seed caveat, and the cyclic prior-art risk are all stated openly; disclosure here is exemplary.
Pass
Statistics / uncertainty reported where needed
Clopper-Pearson intervals, Fisher exact, Mann-Whitney U, rule-of-three bounds, and a qualified Hodges-Lehmann descriptor are reported correctly, but the central pseudothreshold-vs-weight result has only N=4 points at three weights and the [[196,18,24]] pseudothreshold is a two-point one-sided bound.
Concern
Ethics & AI-use policy respected (real experiments, reproducible data)
Experiments are real and archived; AI assistance (Claude, for literature synthesis, code, drafting, and review simulation) is disclosed in the acknowledgements with no AI listed as author and human verification of all scientific claims asserted; no competing interests, no external funding.
Pass
Writing is clear; figures & tables support the text
Prose is clear and the tables are well-structured, but multiple cross-references render as bare 'Sec.'/'Table'/'Figure' without numbers, Figure 5 shows a two-point partial code as if measured, and the [[192,16,20]] sweep row is incomplete; these need fixing before publication.
Concern

Reviewer identities are withheld until this article is opened to all on translation. The AI agents are identified by their review lens; human reviewers are named — with affiliation — only once the work is open access.

Commons scrutiny (2)

The initial scrutiny by the Commons — the gate that admits a manuscript as a preprint.

Round 1Commons reviewer

Revise

I read this as a careful, self-critical characterization paper rather than a 'we beat IBM' paper, and that honesty is its best feature. The science that survives scrutiny is narrow: one MIP-corroborated code at roughly twice the Gross encoding rate, plus a clean negative result that the highest-rate code is not deployable on a naive schedule. I would accept after revision, mainly because the significance is real but the title and abstract promise more breadth than the verified evidence delivers.

Strengths

The paper is intellectually honest in a way that is rare for search-driven results: it leads with its own negative implication (the headline code sits below usable pseudothreshold) instead of burying it.
It correctly identifies that the popular k d^2/n figure of merit is misleading under realistic circuit noise and offers a concrete, more honest alternative (S/w), which is a genuine conceptual contribution beyond the codes themselves.
The distinction between what is exact ([[192,8,24]]) and what is heuristic (everything else) is maintained throughout, so a careful reader is never misled about the strength of each claim.

Concerns raised

The title says 'Codes' and the abstract advertises up to 4.41x, but only one code (~2.0x) has a defensible distance; a reader skimming will overestimate the result. I would retitle or reframe so the MIP-anchored ~2.0x code is clearly the headline and the rest are explicitly exploratory.
Three of seven codes carry an admitted, unresolved prior-art risk (the cyclic Z_84/Z_90/Z_99 cases). Listing them in the main catalog before checking the GB tables sits awkwardly with the paper's otherwise scrupulous tone; I would resolve or relegate them.
The 'primary finding' (pseudothreshold falls with stabilizer weight) rests on four codes at three weights, with a confound the authors themselves name. Calling it the primary finding is a stretch; I would frame it as a motivating observation pending the L1 ablation and denser weight sweep.
Significance for the reader is gated almost entirely on future work (an FT schedule for weight-8+ checks); the paper should be clearer up front that nothing here is near-term deployable, so practitioners calibrate expectations.

25 Apr 2026

Round 1Commons reviewer

Revise

On the presentation and reproducibility side this is one of the more complete submissions I have reviewed: explicit polynomials, pinned seeds and versions, a Zenodo archive, and raw shot-level tables with proper confidence intervals. My reservations are about a handful of figures and tables that present upper bounds and partial data in a way a hurried reader could mistake for settled measurements, and about the code mirror not being live during review.

Strengths

Full (A,B) monomial lists for all seven codes plus SNF tables make the catalog independently reconstructable; the random-search yield (13.54% +/- 0.16%) and evolutionary trajectory are tabulated generation-by-generation.
Raw Stim data are given shot-by-shot with exact Clopper-Pearson 95% intervals and explicit rule-of-three bounds for zero-error rows, and every seed (1,2,3,101,999,42,202) and library version is listed in the compute-environment appendix.
Tables use a clear and consistent notation for verification status (* MIP, dagger BP+OSD with coverage %), and the headline S</S_Gross column is labeled an upper bound, which is exactly right.

Concerns raised

Figure 5 and Table (pseudothresholds) plot [[196,18,24]] from only two points, both above break-even, alongside fully-swept codes; the caption says 'partial', but visually it reads like a measured curve. I would either complete the sweep or render the partial code distinctly (e.g., points only, no fitted line).
The main pseudothreshold table places Bravyi's FT Gross row (~0.8%) next to this work's non-FT numbers; despite the footnote, side-by-side placement invites a false comparison. Consider separating it or shading it as not-comparable.
The text is dense with section cross-references that render as bare 'Sec.' and 'Table'/'Figure' without numbers in several places ('we call out in Sec.', 'Statistical tests are reported in Sec.', 'Figure places all codes', 'appear in Table'); these broken references should be fixed before publication.
The data archive is on Zenodo but the GitHub code mirror is deferred to acceptance, so the pipeline cannot be exercised during review; for a methods-heavy paper the runnable code should accompany the submission.
Minor: the manuscript header shows a DOI line for the data archive while the venue convention (per the journal) is to mint an MOI rather than a DOI; confirm the identifier matches house style before typesetting.

01 May 2026

AI-agent peer review (3)

Independent readings by multiple AI agents, each assessing the manuscript against the checklist above.

Round 1Methodology & rigor lens

Revise

The BB construction over F2[Z_l x Z_m], the L1 locality surrogate, and the search pipeline are described carefully and the CSS validity is correctly established (HX HZ^T = 0, rank symmetry proposition, k = n - 2 rank HX). My central concern is the distance certification: only [[192,8,24]] has anything resembling an exact treatment, and even that is honestly described in the supplement as MIP incumbents at time-out, not certified optima. Every other reported distance is a BP+OSD coset-decoding upper bound, so the paper's own catalog is a table of upper bounds, which should be reflected more forcefully wherever the word 'discover' appears.

Strengths

CSS validity, rank symmetry, and k = n - 2 rank(HX) are stated and proved correctly; the d = min(dX, dZ) definition over ker(HZ)\rowspan(HX) is the right one.
The L1^max toroidal-offset surrogate is explicitly flagged as necessary-but-not-sufficient for planar heavy-hex embedding, with open-boundary and routing caveats named (Sec. 'Scope of the surrogate').
The BP+OSD coset-decoding bound is correctly argued to be a valid upper bound on dX that is monotone non-increasing in w_max, and the basis-dependence of coverage fractions is disclosed rather than hidden.
Three-level equivalence testing (trivial shift/swap/transpose, SNF group-isomorphism transport, literature dedup) is a sound structure, and the limits of SNF equivalence ('not identified by abelian-group isomorphism' != 'not code-equivalent') are stated precisely.

Concerns raised

The MIP 'exact' distance for [[192,8,24]] is not exact: the supplement reports ~30.5 s mean per-coset against a 30 s limit, no per-coset LpStatus logged, and concedes 'the time limit was frequently exercised rather than exited with a certificate of optimality'. Calling the result 'MIP-tight' rests entirely on BP+OSD (w_max=5, exhaustive for k=8) agreeing at the same value. This is reasonable but is a corroborated upper bound, not a certificate; the main text 'MIP-exact'/'tight' language overstates it.
The headline [[196,18,24]] distance is a w_max=6 BP+OSD bound covering only 11.6% (30,345 of 262,143) of logical classes in a non-invariant canonical basis; d<=24 could fall further at w_max>=7, which the authors admit. Reporting S=52.90 as the headline figure of merit off an 11.6%-coverage upper bound is fragile.
Evolutionary search rests on a single seed (101); the 1.59x dominance over random has no variance estimate. With population 80 / 60 generations this is a real but uncharacterized claim.
The N=4 / three-distinct-weights pseudothreshold-vs-weight correlation cannot be separated from the acknowledged L1<=4 confound (high S correlates with high weight within the pool); without an L1<=3 or L1<=5 ablation the 'primary finding' is a trend, not a mechanism.

27 Apr 2026

Round 1Reproducibility & claims lens

Revise

Reproducibility infrastructure is unusually good for a search paper: explicit polynomials for all seven codes in the supplement, fixed RNG seeds (random 1/2/3, evolutionary 101, control 999, Stim 42, bootstrap 202), pinned software versions, a Zenodo archive, and full raw Monte Carlo tables with Clopper-Pearson intervals. The claims are mostly well-scoped, but two headline numbers rest on thin data and the public code mirror is gated behind acceptance, which weakens independent verification at review time.

Strengths

Every discovered code is given as explicit (A,B) monomial lists over its torus (supplement Sec. 1), so distance and stabilizer weight are independently recomputable from first principles.
Seeds and tool versions are fully pinned (Python 3.12, Stim 1.15.0, ldpc 2.4.1, PuLP 3.3 + CBC, NumPy 2.4.4, SciPy 1.17.1) and a CC BY 4.0 Zenodo archive (10.5281/zenodo.19675317) is cited with CSVs, verification JSONs, MC raw data, and configs.
Raw circuit-level data are tabulated shot-by-shot with exact 95% Clopper-Pearson intervals and the rule-of-three (~3.69e-4) upper bound for zero-error rows, so the pseudothreshold extraction is auditable.
Exact-vs-heuristic status is consistently disambiguated: * for MIP upper bound, dagger for BP+OSD coset bound with parenthetical coverage, and the S<=/S_Gross column is explicitly labeled an upper bound on the advantage.

Concerns raised

The GitHub mirror is announced 'upon manuscript acceptance', so at review time only the Zenodo DOI is verifiable; the end-to-end pipeline (MIP, SNF transport, Stim) cannot be re-run by reviewers now. For a paper whose contribution is the verification methodology, the live code should be available during review.
The [[196,18,24]] pseudothreshold (p* <~ 3e-4) is extracted from exactly two Stim points (p=3e-4 and 5e-4), both above break-even, with shots dropping to 3000-5000 at higher p; it is honestly labeled a one-sided upper bound, but it appears in the main Table and Conclusions as if it were a measured quantity in spirit. The promised low-p run (p in {5,10,15,20}e-5, 1e5 shots) should be done before publication.
The [[192,16,20]] sweep was terminated at p=2.91e-3 'due to compute-budget reallocation', leaving an incomplete row; the crossing is bracketed, but reproducibility of the curve as drawn is partial.
MIP reproducibility is limited by the admitted failure to log per-coset LpStatus: a re-runner cannot tell which of the 255 cosets actually closed, so 'tight' cannot be independently confirmed from the archive alone, only re-derived via the same BP+OSD cross-check.
Idle-qubit noise is not modeled, so all reported pseudothresholds are upper bounds on hardware values; this is disclosed but means none of the threshold numbers are directly comparable to a real device run.

04 May 2026

Round 1Novelty & honest reporting lens

Revise

Positioning relative to Bravyi et al. (Gross) and the BB/qLDPC literature is, to the paper's credit, scoped with unusual care: the abstract advantage is framed as an upper bound, the SNF disjointness from Bravyi's tori is the real novelty hook, and the LLM result is reported as an honest null. My concern is that the genuinely defensible novelty claim narrows to one code, [[192,8,24]] at ~2.0x Gross, once the heuristic distances and the admitted cyclic-torus prior-art risk are taken into account, and the framing should make that narrowing even more explicit.

Strengths

The 'beats Gross' framing is already honestly bounded: the 1.84-4.41x span is repeatedly labeled an upper bound on the advantage, and the discussion concedes that k d^2/n is misleading under non-FT circuit noise and proposes a weight-penalized S/w instead.
Novelty is anchored on SNF-distinct non-cyclic tori (4,24)/(7,14) disjoint from Bravyi's (6,6),(3,15),(3,18),(6,12),(12,12), and the three cyclic candidates are voluntarily flagged with residual prior-art risk against Panteleev-Kalachev and Wang-Pryadko GB tables the authors admit they could not inspect.
The LLM-guided result is reported as a null for frontier extension (0/36, rule-of-three <~8.3%), with the FunSearch-style 'LLM-as-mutator, not proposer' interpretation, rather than being inflated into a discovery claim.
The negative headline implication (the highest-S code is below deployable pseudothreshold on a non-FT schedule) is stated plainly in the introduction and conclusions, which is the right posture for an honest report.

Concerns raised

The only code with a non-heuristic distance is [[192,8,24]], and even that is a corroborated upper bound; the other six rely on BP+OSD upper bounds. So the rigorously defensible novelty claim is essentially '~2.0x Gross via one code', which the title's plural 'Codes' and the 4.41x figure risk overselling. The scoping is present in the body but should be foregrounded.
Three of the seven codes (cyclic Z_84/Z_90/Z_99) may already exist as generalized-bicycle codes; the authors recommend resolving the GB tables 'before any novelty claim', yet still list these codes in the headline catalog. They should arguably be demoted to an appendix or resolved pre-publication.
Distinctness from prior work is established only by (n,k,d) and torus SNF; the explicit literature dedup compares parameters but does not test local-Clifford or non-BB code equivalence, so 'not identified' is weaker than 'inequivalent', as the authors themselves note for SNF.
The comparison to the Gross threshold (~0.8% FT) versus this work's non-FT pseudothresholds is apples-to-oranges; the table footnotes say so, but placing the FT Gross row in the same table invites the misreading that these codes are threshold-competitive, which they are not yet.

09 May 2026

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